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On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\mathbb R)$ (1609.04712v1)
Published 15 Sep 2016 in math.AP
Abstract: We prove global existence for the one-dimensional cubic non-linear Schr\"odinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega and Hyakuna-Tsutsumi to the modulation space setting and exploits polynomial growth of the free Schr\"odinger group on modulation spaces.